Skip to main content
SpringerLink
Log in
Book cover

Turbulent Shear Flows 6 pp 68–77Cite as

An Eigenfunction Analysis of Turbulent Thermal Convection

  • Conference paper

Abstract

Turbulent convection is considered for the case of stress-free boundary conditions. The Boussinesq equations are numerically integrated for Pr = 0.72 and Ra = 46 000. This results in a data base which is analyzed by decomposing the flow in terms of eigenfunctions generated from the spatial covariance tensor. It is shown that this leads to a relatively simple dynamical description in terms of relatively few modes. Explicit representations of the major modes and their evolution are presented. As another application it is shown that this approach leads to a substantial compression of the turbulence data.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Abbreviations

e z :

Unit vertical vector

g :

Acceleration of gravity

H :

Height of convection cell

k :

Wavenumber in x-direction

K ij :

Covariance, (10)

L :

Horizontal scale of cell

l :

Wavenumber in y-direction

Nu:

Nusselt number, ratio of heat flow to κΔT/H.

Pr:

Prandtl number (5)

p :

Pressure, departure from equilibrium, normalized by ϱu 2 s

Ra:

Rayleigh number (4)

Ra c :

Free-free critical Ra number = 27 π4/4

R λ :

Taylor microscale Reynolds number = 〈w2〉/(v〈(∂w/∂z)2〉1/2)

T :

Temperature, departure from equilibrium, normalized by ΔT

u = (u, v, w):

Velocity field, normalized by u s

u c :

Characteristic velocity scale (20)

u s :

κ/H Standard velocity scale

v :

= (u, θ) Flow vector

x :

= (x, y, z) Spatial variables

t :

Time

〈 〉:

Denotes ensemble average

α :

Thermal expansivity

δ :

Thermal boundary layer (20)

ΔT :

Temperature difference of the cell

κ :

Thermal diffusivity

λ (n) kl :

Eigenvalue (13)

v :

Kinematic viscosity

ϱ :

Density

θ :

Temperature fluctuation from mean, normalized by ΔT

θ s :

= ΔT Standard temperature scale

θ c :

Characteristic temperature scale (23)

φ (n) kl :

Eigenfunction (13)

References

  1. Lumley, J. L.: “The Structure of Inhomogeneous Turbulent Flows”, in Atmospheric Turbulence and Radio Wave Propagation, ed. by A. M. Yaglom and V. I. Tatarski (Nauka Moskow 1967) pp. 166–178

    Google Scholar 

  2. Lumley, J. L.: Stochastic Tools in Turbulence (Academic, New York 1970)

    MATH  Google Scholar 

  3. Sirovich, L.: Turbulence and the dynamics of coherent structures. Part 1: Coherent structures. Q. Appl. Math. 45/3, 561–571 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Sirovich, L.: Turbulence and the dynamics of coherent structures. Part 2: Symmetries and transformations. Q. Appl. Math. 45/3, 573–582 (1987)

    MathSciNet  Google Scholar 

  5. Sirovich, L.: Turbulence and the dynamics of coherent structure. Part 3: Dynamics and scaling. Q. Appl. Math. 45/3, 583–590 (1987)

    MathSciNet  Google Scholar 

  6. Sirovich, L., Rodriguez, J. D.: Coherent structures and chaos: A model problem. Physics Lett. A 120/5, 211–215 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  7. Sirovich, L., Kirby, M.: Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Am. A 4/3, (1987)

    Google Scholar 

  8. Drazin, P. G., Reid, W. H.: Hydrodynamic Stability (University Press, Cambridge 1981)

    MATH  Google Scholar 

  9. Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods (SIAM, Philadelphia 1977)

    Book  MATH  Google Scholar 

  10. Herring, J. R., Wyngaard: “Convection with a Simple Chemically Reactive Passive Scalar, 10.39–10.43”, Fifth Symposium on Turbulent Shear Flows, Cornell University (1985)

    Google Scholar 

  11. Curry, J. H., Herring, J. R., Orszag, S. A., Loncaric, J.: Order and disorder in two and three dimensional Benard convection. J. Fluid Mech. 147, 1–38 (1984)

    Article  ADS  MATH  Google Scholar 

  12. Deardorff, J. W.: Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211–1213 (1970)

    Article  ADS  Google Scholar 

  13. Sirovich, L., Maxey, M., Tarman, H.: “Analysis of Turbulent Thermal Convection”, in Sixth Symposium on Turbulent Shear Flows, Toulouse (1987)

    Google Scholar 

  14. Gilbert, N., Kleiser, L.: “Low-Resolution Simulations of Transitional and Turbulend Channel Flow”, in Proceedings of the International Conference of Fluid Mechanics, ed. by Shen Yuam (Beijing 1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Applied Mathematics Brown University, Providence, RI, 02912, USA

    L. Sirovich, M. Maxey & H. Tarman

Authors
  1. L. Sirovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Maxey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Tarman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Centre National de Recherches Météorologiques, 42, Avenue Coriolis, 31057, Toulouse Cedex, France

    Jean-Claude André

  2. O.N.E.R.A./C.E.R.T., 2, Avenue Edouard Belin, 31055, Toulouse Cedex, France

    Jean Cousteix

  3. Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, Egerlandstraße 13, 8520, Erlangen, Fed. Rep. of Germany

    Franz Durst

  4. Department of Mechanical Engineering, University of Manchester, Institute of Science and Technology, PO Box 88, Manchester, M60 1QD, England

    Brian E. Launder

  5. Mechanical Engineering Department, The Pennsylvania State University, University Park, PA, 16802, USA

    Frank W. Schmidt

  6. Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London, SW7 2BX, England

    James H. Whitelaw

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sirovich, L., Maxey, M., Tarman, H. (1989). An Eigenfunction Analysis of Turbulent Thermal Convection. In: André, JC., Cousteix, J., Durst, F., Launder, B.E., Schmidt, F.W., Whitelaw, J.H. (eds) Turbulent Shear Flows 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73948-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-73948-4_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-73950-7

  • Online ISBN: 978-3-642-73948-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation